About Alligation or Mixture
Alligation is a shortcut for mixing problems. When two ingredients of different prices or strengths are combined, the value of the mixture lies between the two. The quantities used are in the ratio of the two differences from that middle value, which turns what looks like an equation into a single subtraction.
What you need to understand
- The value of a mixture always lies between the values of the two ingredients being mixed.
- The cheaper ingredient must be taken in proportion to how far the dearer one lies above the mean.
- Quantity of cheaper : quantity of dearer = (dearer - mean) : (mean - cheaper).
- The rule is the same whether the ingredients are prices, concentrations, speeds or averages, because the underlying relation does not change.
- Removal and replacement is a different idea. Each withdrawal takes out a proportion of whatever is present, so repeated operations multiply the fraction remaining instead of subtracting a fixed amount.
- Adding water or another free ingredient dilutes by spreading a fixed total value over a larger volume.
Formulas to remember
How to work through these questions
- Identify the two ingredient values and the value of the finished mixture.
- Write each difference from the mean, then place the dearer difference first and the cheaper difference second.
- Check that the ratio gives more of the cheaper ingredient, because it must.
- For removal and replacement, find the fraction left after one operation and raise it to the number of operations.
- For dilution with water, remember that water adds no value, so the total value is unchanged while the volume grows.
Mistakes that cost marks
- Inverting the alligation ratio and pairing each difference with the wrong ingredient.
- Using the two prices themselves as the ratio instead of the two differences from the mean.
- Subtracting the total quantity drawn off in a replacement question, which ignores that later withdrawals remove a diluted mixture.
- Using the volume left after removal as the denominator instead of the original volume.
- Adding the two differences together to form one term of the ratio.
Worked example
In what ratio must tea costing Rs. 240 per kg be mixed with tea costing Rs. 300 per kg so that the mixture is worth Rs. 260 per kg?
- The cheaper tea lies below the mean by 260 - 240 = 20.
- The dearer tea lies above the mean by 300 - 260 = 40.
- Quantity of cheaper : quantity of dearer = 40 : 20 = 2 : 1.
- Check: two kilograms at Rs. 240 and one at Rs. 300 cost 480 + 300 = Rs. 780 for 3 kg, which is Rs. 260 per kg.
Answer: 2 : 1
Practice questions with answers
A few Alligation or Mixture questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
In what ratio must rice costing Rs. 30 per kg be mixed with rice costing Rs. 40 per kg so that the mixture costs Rs. 34 per kg?
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A
3 : 5
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B
2 : 3
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C
5 : 2
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D
3 : 2
Answer: Option D — with explanation
By the rule of alligation the two quantities are in the ratio of the two differences from the mean price.
The cheaper rice is below the mean by 34 - 30 = 4, and the dearer is above it by 40 - 34 = 6.
Quantity of cheaper : quantity of dearer = 6 : 4 = 3 : 2.
Common mistakes
- used the sum of the differences as the second term: 3 : 5
- inverted the ratio, pairing each difference with the wrong rice: 2 : 3
- added the two differences when forming the first term: 5 : 2
Question 2
A vessel contains 144 litres of pure milk. 24 litres of the milk are drawn out and replaced with water. If this process is repeated 2 times, how much milk is left in the vessel?
Answer: Option B — with explanation
Each time, the milk is diluted by the factor (144 - 24) / 144 = 120/144.
After 2 replacements, milk left = 144 x \(\frac{120}{144}\)^ 2 = 100 litres.
Common mistakes
- used the volume left after removal as the denominator: 92.16
- subtracted the total quantity drawn off as though the mixture were never diluted: 96
- allowed for only one replacement: 120
Question 3
How many litres of water must be added to 7 litres of milk costing Rs. 40 per litre so that the mixture costs Rs. 28 per litre?
Answer: Option C — with explanation
Water costs nothing, so the value of the milk is simply spread over more litres.
The milk is worth 7 x 40 = 280 rupees in all, and this must be spread over (7 + w) litres at Rs. 28 a litre.
So (7 + w) x 28 = 280, giving w = 3 litres.
Common mistakes
- used the original price in the denominator: 2.1
- subtracted the water instead of adding it: 4
- divided by the price difference the wrong way round: 16.33
Question 4
Rice at Rs. 47 per kg is to be mixed with rice at Rs. 56 per kg. In what ratio should they be mixed to get a mixture worth Rs. 51 per kg?
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A
9 : 4
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B
5 : 4
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C
4 : 5
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D
47 : 56
Answer: Option B — with explanation
By the rule of alligation the two quantities are in the ratio of the two differences from the mean price.
The cheaper rice is below the mean by 51 - 47 = 4, and the dearer is above it by 56 - 51 = 5.
Quantity of cheaper : quantity of dearer = 5 : 4 = 5 : 4.
Common mistakes
- added the two differences when forming the first term: 9 : 4
- used the two prices themselves as the ratio: 47 : 56
- inverted the ratio, pairing each difference with the wrong rice: 4 : 5
Question 5
A vessel is full of milk and holds 128 litres. 32 litres are drawn off and the vessel is filled up with water. After this has been done 2 times, how much of the original milk is left?
Answer: Option A — with explanation
Each time, the milk is diluted by the factor (128 - 32) / 128 = 96/128.
After 2 replacements, milk left = 128 x \(\frac{96}{128}\)^ 2 = 72 litres.
Common mistakes
- allowed for only one replacement: 96
- subtracted the total quantity drawn off as though the mixture were never diluted: 64
- used the volume left after removal as the denominator: 56.89
Frequently asked questions
Why is the ratio built from the differences rather than the prices?
Because the cheaper ingredient has to make up its shortfall against the mean, and the dearer one has to offset its excess. The quantities are therefore inversely related to the distances from the mean.
How is alligation different from taking an average?
An average gives both ingredients the same weight. Alligation tells you which weights are needed to reach one particular mixture value.
How much milk is left after repeated replacements?
Multiply the original quantity by the fraction remaining after one operation, raised to the number of operations. After two operations that each remove one fifth, the fraction left is (4/5) x (4/5) = 16/25.
What if the required mixture price equals one of the ingredient prices?
Then the other ingredient would be used in zero quantity, so no mixing is needed. Alligation still gives the right answer, with one term of the ratio equal to zero.
